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Proof step 2 from proof modmod-add2-negpos-pos-01 (previous step)

i + i0 % j < 0, i < 0, 0 < j, 0 ≤ i + i0, 0 ≤ i0 ⊦ (i + i0 % j) % j = (i + i0) % j - j, (i + i0 % j) % j = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
i + i0 % j < 0, i < 0, 0 < j, 0 < i0, 0 ≤ i + i0, 0 ≤ i0, j ≤ i0 ⊦ (i + i0) % j = (i + i0 % j) % j + j, (i + i0 % j) % j = 0, j = 1

Rule argument

Additional information

Used simplifier rules: mod-ls-same-01: j ≠ 0 ⊦ i % j < i ↔ 0 < i ∧ (abs(j) = 1 ∨ abs(j) ≤ i);%mod-one: ⊦ i % 1 = 0;%mod-zero-09: ⊦ 0 < i ∧ i < j → i % j = i;%forward-le: ⊦ i ≤ j ∧ 0 < i → 0 < j; (forward) from specification int-abs% ⊦ i ≤ j → ¬ j < i; (s)% ⊦ i = i + j ↔ j = 0; (s)% ⊦ i ≤ j ↔ ¬ j < i; (ss)% ⊦ 0 < j → abs(j) = j; (s)% ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2% ⊦ i + 0 = i; (s)% ⊦ i + j < i + i0 ↔ j < i0; (s)%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs% ⊦ i - j = i0 ↔ i = i0 + j; (s)